[EDIT: the text below is correct, but a bit too snarky. My apologies.
Also, the site owner could just trust the client to actually run his/her bitcoin-generating code for a couple of hours and send any generated coins - this should earn him/her ~$0.20/hour, rounded down because lots of people have crufty computers. Still not feasible, and generating bitcoins will only get harder in the future, but not as bad as I suggest below.]
Sure, if you're willing to leave your browser consuming 100% CPU for a couple of weeks. And if you're willing to pay the entire resulting 50 bitcoins (~$40-45) you generated (at the cost of ~$60 in electricty, I'd guess)[1], or trust the site to return you the rest. In either case, you'd need to trust the site code to tell you that it's generated a coin, instead of silently sending it back to the server and continuing to run.
At least, that's my understanding of how bitcoins work.
[1] Only GPU-based generation is currently profitable, and there's no way NaCl can be given full access to the GPU - GPUs typically have DMA access, so being able to run arbitrary code on them is more or less equivalent to kernel-level access. In theory, you could verify the GPU code - but that's an enormous job, and there are lots of GPUs that you'd need to check for security holes. And security was not design criterion #1 for these devices, I'd guess.
Something like DirectX/OpenGL access makes sense, but running crypto algorithms through OpenGL (instead of CUDA or the like) is probably infeasible.
I like that idea. It seems like microgeneration is a feature Bitcoin is currently suffering from a lack of. As competition increases, generation seems to be more and more centralized. A low-traffic way to generate small amounts of Bitcoin in a peer-to-peer network would be a great addition to a future Bitcoin protocol.
This doesn't seem like a good match. With different hardware and prices of electricity out there it would be hard to make the system fair for everyone. Also, although minting coins is an important feature of Bitcoin, it seems there's a certain amount of miners needed to offset threat of fraudulent transactions; going significantly above that amount is just wasting planet's resources.
